Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | D3×#124 |
| Fitting | C3×#124 ⋊ C2 |
| Sylow | C3 ⋊ C2×#124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D3 |
| commutator, socle, cosocle | C22×C3 | C24 |
| Fitting | C3×#124 | C2 |
| prime radical | C2×D3×D4 | C2 |
| hypercenter | #124 | D3 |
| hypocenter | C3 | C2×#124 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #124 | |
330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3)(4,6,7,5)(9,13)(12,15);(4,5,7,6)(8,10,11,14)(9,12,13,15);(5,6)(8,9)(10,12)(11,13)(14,15);(2,3)42 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#329, #68343, #68347, … |
| maximal normal subgroup of | D3×#329, #68343, #68347, … |
| maximal quotient of | #74149, #74164, #74167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1232);